金融高频数据计量——理论与实证(三)

本篇为市场微观结构下篇,主要对市场微观结构的研究方法和方向做一个简要的介绍。更加深入和全面的研究可以参考 O’Hara (1995), Madhavan (2000), Biais et al. (2005)Hasbrouck (2007)

市场微观结构 (market microstructure) 理论主要关注的是投资者潜在的需求是如何被最终反映在价格和交易量上的这一过程。所以,市场微观机构理论的核心主题就是关于价格形成 (price formation)、价格发现 (price discovery)、库存 (inventory)、流动性 (liquidity)、交易成本 (transaction cost)、信息扩散与传播 (information diffusion and dissemination)。传统的市场微观结构理论在解释价格行为上分为两大方向:信息不对称模型 (asymmetric information based model) 和库存模型 (inventory model)。前一模型从信息不对称和逆向选择的视角对市场的动态和价格的修正过程进行建模,有顺序交易模型 (sequential trade model) 和策略交易模型 (strategic trade model) 两大主要研究方向。库存模型则主要研究订单流的不确定性,以及流动性提供商的库存风险,和在一定的风险厌恶下的优化问题。最后我们还会对关于交易行为、订单簿均衡市场 (equilibria limit order book market) 的最近的一些研究成果做一个简要的介绍。

1. 信息不对称模型

1.1 顺序交易模型

在顺序交易模型中,随机的交易者会依次到达市场中。这一框架的是基于这一个假设:市场中的交易者拥有信息的程度不同,并依此划分为知情交易者 (“informed traders”),他们根据自己拥有的关于资产价值的信息来交易;和流动性交易者 (“liquidity traders”),他们因为外在的驱动交易,比如资产组合的调整或者流动性原因等等。根据这一 对交易者不同类型划分的假设派生了大量信息不对称模型,其中开创性的论文包括 Copeland and Galai (1983)Glosten and Milgrom (1985)

Glosten and Milgrom (1985) 的模型中,证券有一个支付 (payoff, 这个payoff与期权定价模型中的payoff含义是一样的,可以粗略的理解为回报,但并不准确),以某个概率可能高也可能低,只有在交易结束后才能最终知道。所有交易者被分为知情交易者,和不知情交易者,知情交易者拥有证券支付的信息,他们会在支付高的情况下买入证券,支付低的情况下卖出证券,而不知情交易者等概率地买入或者卖出证券。市场中的知情交易者的比例是给定的。交易商属于不知情交易者,他们根据交易历史来推断资产的真实价值。当交易商收到一个买入或者卖出的订单后,他会计算这时资产价值的条件期望,然后根据这一条件期望来报出买价和卖价,从而使得自己的期望收益能够抵消与知情交易者交易产生的损失。当下一交易完成后,交易商会再重新进行一次上面的操作,这实际上就是一个适应过程 (adoption process)。按照这个过程,买卖价差变成了资产潜在价值及其相对应的概率(或者两者合起来是数学期望)和市场中知情交易者的比例这三个变量的函数。这一模型的基本假设包括:交易价格是一个鞅;订单流是自相关的(买(卖)单之后倾向于还是买(卖)单),随着时间的推移,交易商的不确定在减少,买卖价差也随之缩小;每笔交易都有价格冲击。

这一模型随后被扩展和修改成很多版本,Easley and O‘Hara (1992) 假设在每个交易日的开始是否有“有价值的信息”出现是随机的,在没有信息出现的情况下,知情交易者并不会交易,只有不知情交易会参与交易;他们还假设不知情交易者也不必一直交易,不知情交易者也可能不参与交易。所以没有交易的间隔,同样informative。其他版本的模型还包 Easley
and O’Hara (1987)
的模型中允许订单的大小可以不同,Easley and O‘Hara (1991) 的模型还考虑了不同类型的订单。

Easley et al. (1997)Easley et al. (2002) 扩展了 Easley and O‘Hara (1992) 的模型,影响资产真实价值的事件是按照泊松过程到达的,这样交易者的顺序到达的时间就不必是离散时间而是连续时间了。知情交易者和不知情交易者的到达过程是两个速率不同的泊松过程,每天买和卖的交易次数的联合分布就变成了混合的泊松分布,那么基于事件到达的速率、知情交易者的到达速率、非知情交易者的到达速率,知情交易者概率 PINprobability of informed trading, 随机选定一个交易者是知情交易者的概率)就可以算出来了。Easley et al. (2008) 又将模型扩展为动态(非齐次泊松过程),用以估计时变 (time-varing) PIN 。

1.2 策略交易模型

在顺序交易模型中,每个交易者只会参与市场一次,所以他们不会考虑他的交易会对后续的交易者产生什么样的影响,知情交易者总是会一次性交易所有数量,因为他们不必考虑将来交易中可能出现的不利价格。在策略交易模型中则不是这样,交易者需要重复参与到市场中,所以也必须依照策略行动。在这个领域的开创性的工作是 Kyle (1985)。Kyle模型中,资产的真实价格是随机的,但是知情交易者知晓资产的真实价格,不知情交易者独立于资产真实价格提交随机的订单(噪声交易,noise trading)。做市商会收到知情交易者和不值钱交易者的订单请求,需要据此报价使得所有交易的盈亏互相抵消。知情交易者可能会采用很激进的交易策略,所以做市商为了防止自己在市场趋势的对立面,需要将报价设置为总净订单量(知情交易者和非知情交易者的所有买单减去卖单后的申买量) 的线性函数。这个过程也需要知情交易者的参与,他们根据对做市商价格设置的猜测和自己的需求(申买/卖量)来计算自己的利润。需要注意的是,与顺序交易模型不同,不知情交易者的交易可能会导致做市商的价格设置使得知情交易者出现亏损。知情交易者的最优下单策略问题变成了选择一个最优的需求使得他的期望利润最大,需求就变成了资产真实价值的线性函数。

当做市商猜测知情交易者的优化问题时,他可以依据自己的价格设置规则来计算知情交易者的需求函数。这样知情交易者的需求函数又会反过来影响做市商的价格设置。做市商价格设置的斜率(即净订单量对价格的影响)又被称作为 “Kyle’s lambda”。

Kyle利用二元正态变量的性质,使用总需求与资产实际价格间的协方差和噪声交易的方差以及他们的函数进行了数值计算。Kyle模型的一个结果是,知情交易者的期望利润随着资产实际价格与做市商无条件报价(与总净订单量无关的报价)的发散,以及噪声交易方差的扩大而增加。后面一个结果表明知情交易者在流动性更好的市场中通常会获得更好的收益。

Kyle模型也被扩展成很多不同的模型,比如 Admati and Pfleiderer (1988)Foster and Viswanathan (1990) 允许不知情交易者也有自己的策略,Foster and Viswanathan (1996), 允许多期模型。

2. 库存模型

库存模型主要考虑在面对异步到达的买方和卖方所造成的库存问题。最初由 Garman (1976) 提出,他用泊松过程对买方和卖方的到达进行建模,他们的到达速率取决于价格。所以当两者速率相同时,做市商处于动态平衡之中。做市商通过买卖价差赚取利润,当买卖价差增大时,单次交易的利润扩大,但是交易的速率下降。 Garman (1976) 将库存问题描述为做市商必须维持所持证券和现金不低于特定水平。如果买卖报价的设置使得买方和卖方的到达速率相等,那么所持证券的库存是一个无漂移的随机游动 (zero-drift random walk),所持现金是正漂移的随机游动(只要买卖价差是正的)。这会导致做市商所持证券会以概率1破产,因为对无漂移的随机游动以概率1到达任意有限水平。因此,只要做市商保持买价和卖价不变,那么他(所持证券)迟早要破产。Amihud and Mendelson (1980) 提出了一个类似的模型,做市商的库存被限制在一个上下限之内。他们的模型表明,当库存水平达到临界值时需要更新报价,从而相应地增加或者降低买方和卖方的到达速率。从而买卖报价随着库存水平的增加单调递减,同时买卖报价并不一定要对称地分布在资产的真实价值两侧。

做市商价格的设置也可以在做市商风险厌恶,希望通过买卖报价适当地平衡自己的资产这一框架下来分析。这些工作包括 Stoll (1978), Ho and Stoll (1981), Stoll (1989)Huang and Stoll (1997) 等等。

3. 交易变量的主要结论

关于微观市场变量的一些主要结论:

交易量:在 Easley and O’Hara (1987) 的模型中,交易者可以选择订单的大小,但是不能不参与交易,订单的大小表明了市场信息的存在。Blume et al. (1994) 研究了交易者在不同时间收到不同质量的信息会对交易量产生什么影响,他分析了交易量与市场价格相关的统计性质,表明交易者可以通过交易量来推断市场中存在的信息的数量与质量。一个重要的结果是,交易量提供的额外信息是无法单纯通过价格推断出来的,交易量和波动性是相关联的。

买卖价差:Glosten and Milgrom (1985) 的模型中,做市商通过对决定买卖价差来补偿自己所面对的逆向选择的风险。做市商与知情交易者交易所产生的潜在损失越高,买卖价差越大。此外,买卖价差与做市商的库存风险与风险厌恶程度相关。在 Easley and O‘Hara (1992) 的模型中,做市商通过交易间时间间隔来推断是否新的信息。因此,滞后的间隔时间和买卖价差是负相关的 (lagged durations and the size of spreads are negatively correlated),因为当间隔时间变小时代表了新信息的出现,做市商的风险上升,买卖价差要相应增加。

交易间隔:Diamond and Verrecchia (1987) 提出了一个合理期望模型,模型对做空进行了限制。他们认为交易的缺失是由于“坏”消息的出现,所以交易的缺失提供了有用的信息并且与价格的波动性相关。在这一模型中,由于对做空进行了限制,所以时间间隔是有信息的。在 Easley and O‘Hara (1992) 模型中,当出现信息时,知情交易者会进入市场,没有得到信息的交易者并不会交易,所以较短的交易间间隔表明了信息的出现。Admati and Pfleiderer (1988) 为交易间隔的在时间轴的聚集提供了一种解释,在他们的模型中,流动性交易者倾向于最小化自己的交易成本,在其他交易者在市场中的时候进行交易。在均衡状态下,对知情交易者来说,知情交易者的行为都相似的时候是最优的。这就导致了交易会聚集,交易间隔是正自相关的。

4. 订单簿模型 (Models for Limit Order Book Markets)

A seminal paper to model limit order markets is Glosten (1994). In this model, all market participants have access to an electronic screen. Posting limit orders is done costlessly and the execution of a trade against the book occurs in a “discriminatory” fashion. That is, each limit order transacts at its limit price. Investors are rational and risk averse and maximize a quasi-concave utility function of their cash and share position as well as personal preferences. The trading behavior of market order traders depends on their marginal valuation functions and the prevailing terms of trade, i.e., the list of bid and ask quotes available, which influence the changes in investors’ cash and share positions. It is assumed that an investor chooses the trade quantity such that her marginal valuation equals the marginal price corresponding to the price paid for the last share in a transaction. There is informed trading if an investor’s marginal valuation is associated with the future payoff. Then, incoming market orders reveal information about the unknown “full information value” of the traded security. Due to the anonymity of the electronic market, the underlying marginal valuation implied by an arriving market order can be assessed by the liquidity suppliers only through the observed limit price and the traded quantity given the terms of trades offered by the book.

Glosten assumes that there is a large number of uninformed, risk-neutral and profit-maximizing limit order submitters who set limit prices and quantities on the basis of their “upper tail expectation”. The latter corresponds to the conditional expectation of the asset’s full information liquidation value given that the next arrival’s marginal valuation is greater than or equal to the traded quantity. In the presence of private information, liquidity suppliers protect themselves against adverse selection by setting the limit price at least equal to the upper tail expectation given a market order trading at the corresponding price. It is shown that such a strategy leads to a Nash equilibrium which is characterized by a zero-profit condition for prices at which positive quantities are offered.

Glosten’s model is extended in several directions by Chakravarty and Holden (1995), Handa and Schwartz (1996), Seppi (1997), Kavajecz (1999), Viswanathan and Wang (2002) and Parlour and Seppi (2003). However, while static equilibrium models provide insights into the structure of the limit order book, they do not allow to analyze (dynamic) interactions between the order flow and the state of the limit order book. For this reason, Parlour (1998) proposes a dynamic game theoretical equilibrium model where traders have different valuations for the asset and choose between submitting a market order, a limit order or refraining from trading. Since the expected future order flow is affected by their own order submission strategies, the execution probabilities of limit orders are endogenous. This leads to systematic patterns in traders’ order submission strategies even when there is no asymmetric information in the market. The basic underlying mechanism is a “crowding out” effect whereby market orders and limit orders on the individual market sides crowd out one another when the ask or bid queue is changed. In particular, the probability of the arrival of a buy (sell) trade after observing a buy (sell) trade is higher than after observing a sell (buy) trade. This results from a buy transaction reducing the depth on the ask side which in turn increases the execution probability for limit sell orders. Hence, for a potential seller, the ttractiveness of limit orders relative to market orders rises inducing a crowding out of market sell orders in favor of limit sell orders. Handa et al. 2003 extend this approach by introducing an adverse selection component due to the presence of privately informed traders.

An alternative dynamic game theoretical equilibrium model has been proposed by Foucault (1999) in order to study the cross-sectional behavior of the mix between market orders and limit orders and the implied trading costs. He analyzes the influence of the risks of being picked off and of non-execution on traders’ order submission strategy and derives testable implications regarding the relationship between the proportion of limit orders and market orders in the order flow, the fill rate (i.e., the percentage of executed limit orders), the trading costs and the volatility of the asset price. Handa et al. (2003) extend the approach by Foucault (1999) by introducing private information in his model. While in Foucault’s model trading occurs because of differences in traders’ valuation for the security, Handa et al. introduce an adverse selection component due to the presence of privately informed traders. As a result, the size of the spread is a function of the differences in valuation among investors and of adverse selection. Further extensions of these frameworks are, among others, oucault et al. (2005) and Goettler et al. (2005, 2009). Recent literature focuses on the theoretical analysis on automated trading and smart order routing in electronic trading platforms. See, e.g., Foucault and Menkveld (2008), Hendershott et al. (2011) or Biais et al. (2010), among others.

最后一部分我并没有看的十分明白,为了严谨起见,我暂时没有翻译,将原文直接贴出来,欢迎交流!
这一篇我在查阅相关文献的过程中,能明显的感到到这一篇的信息量很大,首先参考文献的数量很多,其次引用的出处大都为JF、JFE这样的顶级期刊,而且按照 google scholar 上的引用数据,文章动辄引用量都是几百,还有很多篇引用量上千。有志于这方面研究的朋友,这篇文章是个不错的索引。


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